Based on two-grid discretizations, in this paper, some new local and
parallel finite element algorithms are proposed and analyzed for the
stationary incompressible Navier-Stokes problem.
These algorithms are motivated by the observation that
for a solution to the Navier-Stokes problem, low frequency
components can be approximated well by a relatively coarse grid and
high frequency components can be computed on a fine grid by some
local and parallel procedure. One major technical tool for the
analysis is some local a priori error estimates that are also
obtained in this paper for the finite element solutions on general
shape-regular grids.